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Real-Life Applications of Differential Equations in Population Modeling and Cooling Scenarios

Learn how to apply differential equations to model exponential growth and decay, Newton’s Law of Cooling, and the logistic equation for population dynamics. Solve practical problems involving population growth, disease spread, and thermal cooling scenarios using mathematical models. Understand key assumptions and realistic growth dynamics in population modeling to make accurate predictions and solve complex scenarios. Explore the logistic equation for constrained population growth and its real-world implications.

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Real-Life Applications of Differential Equations in Population Modeling and Cooling Scenarios

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  1. Lesson 57 - Applications of DE Calculus - Santowski Calculus - Santowski

  2. Lesson Objectives • 1. Apply differential equations to applications involving exponential growth & decay, Newton’s Law of Cooling, and introduce the Logistic Equation for population modeling Calculus - Santowski

  3. Fast Five • Solve the following DE: Calculus - Santowski

  4. (A) Exponential Growth • Write a DE for the statement: the rate of growth of a population is directly proportional to the population • Solve this DE: Calculus - Santowski

  5. (A) Exponential Growth • The population of bacteria grown in a culture follows the Law of Natural Growth with a growth rate of 15% per hour. There are 10,000 bacteria after the first hour. • (a) Write an equation for P(t) • (b) How many bacteria will there be in 4 hours? • (c) when will the number of bacteria be 250,000? Calculus - Santowski

  6. (B) Law of Natural Growth • Solve the following problem using DE: • A disease spreads through Doha at a rate proportional to the number of people in Doha. If 50 people are infected on the third day and 146 people are infected on the 6th day, determine • (i) how many people will be infected on day 12? • (ii) Relief from the UN will come only when at least 7.5% of Doha’s population (currently say 850,000) is infected. When will relief come? Calculus - Santowski

  7. Example from IB The population of mosquitoes in a specific area around a lake is controlled by pesticide. The rate of decrease of the number of mosquitoes is proportional to the number of mosquitoes at any time t. Given that the population decreases from 500 000 to 400 000 in a five year period, find the time it takes in years for the population of mosquitoes to decrease by half. Calculus - Santowski

  8. Example from IB • A certain population can be modelled by the differential equation dy/dt = kycoskt, where y is the population at time t hours and k is a positive constant. • (a) Given that y = y0 when t = 0, express y in terms of k, t and y0. • (b) Find the ratio of the minimum size of the population to the maximum size of the population. Calculus - Santowski

  9. (B) Law of Natural Growth • Our previous question makes one KEY assumption about spread of disease (and population growth in general) • Which is ….. ????? • So now we need to adjust for more realistic population growth assumptions Calculus - Santowski

  10. (C) Newton’s Law of Cooling • Write a DE for the following scenario: • The rate at which a hot object cools to the ambient temperature of its surroundings is proportional to the temperature difference between the body and its surroundings, given that T(0) = To • Now solve this DE Calculus - Santowski

  11. (C) Newton’s Law of Cooling • The DE can be written as • Where T(t) is the temperature at any given time and A is the ambient (room) temperature • And the solution turns out to be Calculus - Santowski

  12. (C) Newton’s Law of Cooling • In a steel mill, rod steel at 900ºC is cooled by forced air at a temperature of 20ºC. The temperature of the steel after one second is 400ºC. When will the steel reach a temperature of 40ºC? Calculus - Santowski

  13. (C) Newton’s Law of Cooling • A pie is removed from a 175ºC oven. The room temperature is 24ºC. How long will it take the pie to cool to 37ºC if it cooled 60º in the first minute? Calculus - Santowski

  14. (C) Newton’s Law of Cooling • A thermometer reading -7ºC is brought into a room kept at 23ºC. Half a minute later, the thermometer reads 8ºC. What is the temperature reading after three minutes? Calculus - Santowski

  15. (D) Logistic Equation • A more realistic model for population growth in most circumstances, than the exponential model, is provided by the Logistic Differential Equation. • In this case one’s assumptions about the growth of the population include a maximum size beyond which the population cannot expand. • This may be due to a space limitation, a ceiling on the food supply or the number of people concerned in the case of the spread of a rumor. Calculus - Santowski

  16. (D) Logistic Equation • One then assumes that the growth rate of the population is proportional to number of individuals present, P(t), but that this rate is now constrained by how close the number P(t) is to the maximum size possible for the population, M. • A natural way to include this assumption in your mathematical model is • P’ (t) = k⋅P(t)⋅ (M - P(t)). Calculus - Santowski

  17. (D) Logistic Equation • The differential equation can be interpreted as follows (where M refers to the total population (max if you will) and P is obviously the present number of people infected at time, t): the growth rate of the disease is proportional to the product of the number of people who are infected and the number of people who aren’t Calculus - Santowski

  18. (D) Logistic Equation • So to solve this equation, we separate our variables and try to integrate ….. Calculus - Santowski

  19. (D) Logistic Equation • Now we notice something new in our separable DE  the denominator of the fraction is a product!! • This gets into a technique called partial fractions (AARRGGHHH), so we will avoid it and simply use some tricks on the TI-89 Calculus - Santowski

  20. (D) Logistic Equation • Consider the product • So what two fractions where added together to form this product fraction ==> i.e. Calculus - Santowski

  21. (D) Logistic Equation • Consider the product • So what two fractions where added together to form this product fraction ==> i.e. • You are welcome to run through the algebra yourself, but why not simply expand Calculus - Santowski

  22. (D) Logistic Equation • So back to our DE • We get • So this will simplify our integration because the integral of a sum/difference is simply the sum/difference of integrals ….. Calculus - Santowski

  23. (D) Logistic Equation • So then integrating gives us Calculus - Santowski

  24. (D) Logistic Equation • And upon simplification, we get • Which we call a logistic function • And upon graphing: Calculus - Santowski

  25. (E) Examples • Fish biologists put 200 fish into a lake whose carrying capacity is estimated to be 10,000. The number of fish quadruples in the first year (a) Determine the logistic equation (b) How many years will it take to reach 5000 fish? (c) When there are 5000 fish, fisherman are allowed to catch 20% of the fish. How long will it take to reach a population of 5000 fish again? (d) When there are 7000 fish, another 20% catch is allowed. How long will it take for the population to return to 7000? Calculus - Santowski

  26. Mixing Problems • Ex 1 • Ex 2 • Ex 3 Calculus - Santowski

  27. Mixing Problems – Worked Example • A tank contains 20 kg of salt dissolved in 5000 L of water. • Brine that contains 0.03 kg of salt per liter of water enters the tank at a rate of 25 L/min. • The solution is kept thoroughly mixed and drains from the tank at the same rate. • How much salt remains in the tank after half an hour?

  28. MIXING PROBLEMS • Let y(t) be the amount of salt (in kilograms) after t minutes. • We are given that y(0) = 20 and we want to find y(30). • We do this by finding a differential equation satisfied by y(t).

  29. MIXING PROBLEMS • Note that dy/dt is the rate of change of the amount of salt. • Thus, where: • ‘Rate in’ is the rate at which salt enters the tank. • ‘Rate out’ is the rate at which it leaves the tank.

  30. RATE IN • We have:

  31. MIXING PROBLEMS • The tank always contains 5000 L of liquid. • So, the concentration at time t is y(t)/5000 (measured in kg/L).

  32. RATE OUT • As the brine flows out at a rate of 25 L/min, we have:

  33. MIXING PROBLEMS • Thus, from Equation 5, we get: • Solving this separable differential equation, we obtain:

  34. MIXING PROBLEMS • Since y(0) = 20, we have: –ln 130 = CSo,

  35. MIXING PROBLEMS • Therefore, • y(t) is continuous and y(0) = 20, and the right side is never 0. • We deduce that 150 – y(t) is always positive.

  36. MIXING PROBLEMS • Thus, |150 – y| = 150 – y. • So, • The amount of salt after 30 min is:

  37. MIXING PROBLEMS • Here’s the graph of the function y(t) of Example 6. • Notice that, as time goes by, the amount of salt approaches 150 kg.

  38. Further Mixing Problems Calculus - Santowski

  39. Further Mixing Problems Calculus - Santowski

  40. Further Mixing Problems Calculus - Santowski

  41. Further Mixing Problems Calculus - Santowski

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